
see e.g. [1]}, [2]}, [3]},
where \(\rfloor \)  denotes the so-called Hook operator allowing for
the natural contraction between tensor fields. In (),
the variational derivative \(\delta \mathfrak {H}=\delta _{\alpha }\mathcal {H}\mathrm {d}x^{\alpha }\wedge \mathrm {d}z\) ,
with \(\wedge \)  denoting the exterior (wedge) product, locally reads
as \(\delta _{\alpha }\mathcal {H}=\partial _{\alpha }\mathcal {H}-d_{z}(\partial _{\alpha }^{z}\mathcal {H})\) .
Furthermore, the linear operators \(\mathcal {J},\mathcal {R}:\mathcal {T}^{*}(\mathcal {E})\wedge \mathcal {T}^{*}(\mathcal {B})\rightarrow \mathcal {V}(\mathcal {E})\) 
describe the internal power flow and the dissipation effects of the
system, respectively. The coefficients \(\mathcal {J}^{\alpha \beta }\) 
of the interconnection tensor \(\mathcal {J}\)  meet \(\mathcal {J}^{\alpha \beta }=-\mathcal {J}^{\beta \alpha }\in C^{\infty }(\mathcal {J}^{2}(\mathcal {E}))\) ,
while we have \(\mathcal {R}^{\alpha \beta }=\mathcal {R}^{\beta \alpha }\in C^{\infty }(\mathcal {J}^{2}(\mathcal {E}))\) 
and \([\mathcal {R}^{\alpha \beta }]\ge 0\)  for the coefficient matrix
of the symmetric and positive semi-definite dissipation mapping \(\mathcal {R}\) .
With respect to the dual input and output bundles \(\rho :\mathcal {U}\rightarrow \mathcal {J}^{2}(\mathcal {E})\) 
and \(\varrho :\mathcal {Y}\rightarrow \mathcal {J}^{2}(\mathcal {E})\) ,
we have the input map and its adjoint output map \(\mathcal {G}:\mathcal {U}\rightarrow \mathcal {V}(\mathcal {E})\) 
and \(\mathcal {G}^{*}:\mathcal {T}^{*}(\mathcal {E})\wedge \mathcal {T}^{*}(\mathcal {B})\rightarrow \mathcal {Y}\) ,
respectively, and thus, the relation \((u\rfloor \mathcal {G})\rfloor \delta \mathfrak {H}=u\rfloor (\mathcal {G}^{*}\rfloor \delta \mathfrak {H})=u\rfloor y\) 
holds, see [1]} or [2]}.
To be able to determine the formal change of the Hamiltonian functional
\({H}=\int _{0}^{L}\mathcal {H}\mathrm {d}z\)  along solutions
of (REF ), we make use of the Lie-derivative
\(\mathrm {L}_{j^{1}(v)}\) , where we set \(v=\dot{x}\)  with (REF ),
see [6]}, and thus, we obtain
\(\dot{{H}}=-\int _{0}^{L}\mathcal {R}(\delta \mathfrak {H})\rfloor \delta \mathfrak {H}+\int _{0}^{L}u\rfloor y+\left.(\dot{x}\rfloor \delta ^{\partial }\mathfrak {H})\right|_{0}^{L}\,\) 
